The study of the dynamical behavior of dark energy is often done in terms of its equation of state \(w(z)=\frac{P(z)}{\rho (z)}\)  that can vary as a function of redshift. Equation of state \(w=-1\)  corresponds to the cosmological constant.
There are some recent studies where it has been shown that solving of \(H_0\)  and \(S_8\)  tensions require \(w(z)<-1\)  at some \(z>0\)  and time-varying dark energy equation of state which cross the phantom barrier [1]}. Also it has been shown that a large class of quintessence (\(w > -1\) ) models including the ones which arise from string swampland conjecture lower the \(H_0\)  parameter and thereby makes \(H_0\)  tension worse [2]}.
From observations, it's required that equations of state at present time should be consistent with value \(w\approx -1\) , however, constraints on the equation of state at higher redshift are weaker. There have already been several efforts to parameterize the equation of state of dark energy. Some recent works in this direction can be found in [3]}, [4]}, [5]}, [6]}, [7]}, [8]}, [9]}, [10]}, [11]}, [12]}, [13]}, [14]}, [15]}, [16]}, [17]}, [18]}, [19]}, [20]}, [21]}.
